Compact, continuous embeddings of $W^{1,p} \leftrightarrow C^{(\alpha)}$

Compact, continuous embeddings of $W^{1,p} \leftrightarrow C^{(\alpha)}$

The sobolev-space $H^s([-\pi,\pi])$ can be embedded into
$C^{(\alpha)}([-\pi,\pi])$ (space of $\alpha$-Hölder-continuous functions)
and vice-versa.
My question is for which exponents $s, \alpha$ can we reach those
embeddings and for which exponents $s, \alpha$ are these embeddings
compact?
$H^s \subset C^{(\alpha)} $ continuous for $s > \alpha + [?]$
$H^s \subset C^{(\alpha)} $ compact for $s > \alpha + [?]$
$C^{(\alpha)} \subset H^s $ continuous for $s < \alpha - [?]$
$C^{(\alpha)} \subset H^s $ compact for $s < \alpha - [?]$